Dependencies

Let the entries of \(L\in M_N(\mathbb {R})\) be independent, mean zero, and square-integrable with common variance \(\sigma ^2\). Then

For \(N{\gt}0\) and \(\sigma \ne 0\), the ratios to \(\mathbb {E}\lVert L\rVert ^2\) are \((N+1)/(2N)\) and \((N-1)/(2N)\) respectively. If the entries are iid nondegenerate centered Gaussian, then also

Let \(W_K,W_Q\in M_{n\times N}(\mathbb {R})\) have jointly independent, mean-zero, square-integrable entries, with variances \(\sigma _K^2\) and \(\sigma _Q^2\) respectively. For \(L=W_K^TW_Q\),

For positive dimensions and nonzero variances, the ratios of the last two expectations to the first are \((N+1)/(2N)\) and \((N-1)/(2N)\).

For \(m\ge 1\), let \(\Lambda _m\) consist of the unit vectors \(\lambda _1\le \cdots \le \lambda _{2m}\) with \(\lambda _m{\lt}0{\lt}\lambda _{m+1}\). Set

Define

For a vector \(\lambda \) with \(|\lambda _i|{\lt}1\), define

For \(L\in M_N(\mathbb {R})\), put

Then \(L=S+T\), with \(S\) symmetric and \(T\) antisymmetric.

For \(L\ne 0\), let \(\alpha \) and \(\beta \) be the decreasing lists of positive eigenvalues of \(S\) and absolute values of its negative eigenvalues, padded with zeros to equal length. Define \(\pi (L)=(a,b,c,d)\) by

Positive and negative parts are taken entrywise. Lean pads both lists to length \(N\) and includes the denominator in each definition.

For \(\lambda \in \Lambda _m\),

Moreover, \(0\le 2C\le 1\) and \(0\le \mathcal P\le 1\).

For any \(L\in M_N(\mathbb {R})\),

For \(L\ne 0\), division gives

For \(W_K,W_Q\in M_{n\times N}(\mathbb {R})\), set

The symmetric part of \(W_K^TW_Q\) is \(S=\tfrac 12M^TJM\). For every \(k\ge 1\),

If \(M:\mathbb {R}^N\to \mathbb {R}^{2n}\) is surjective and \(n\ge 1\), the nonzero spectrum of \(S\), with multiplicity, is the spectrum of \(\tfrac 12JG\). Its sorted normalized nonzero spectrum \(\lambda \) exists uniquely and belongs to \(\Lambda _n\). For \(H=JG/(2\lVert S\rVert )\),

The map \(\iota \) is an orthogonal, self-adjoint linear involution, preserves \(\Lambda _m\), and exchanges \(\lambda _+\) and \(\lambda _-\). Consequently \(\mathcal I\circ \iota =-\mathcal I\), \(C\circ \iota =C\) and \(\mathcal P\circ \iota =\mathcal P\). For \(\lambda \in \Lambda _m\),

In particular, \(\mathcal P(\lambda )=1\) if and only if \(\iota \lambda =\lambda \).

Every \(\lambda \in \Lambda _m\) has \(|\lambda _i|{\lt}1\). Writing \(n=2m\),

and both series converge absolutely. Moreover,

If a symmetric matrix \(A\) has spectrum \(\lambda \), with multiplicity, then

Let \(X{\gt}0\) almost surely, with finite nonzero second moment, and fix \(u,v{\gt}0\), \(\rho =u/v\). Let \(\alpha _n\) be the decreasing sorting of \(vX_1,\ldots ,vX_n\) and \(\beta _n\) the decreasing sorting of \(uX'_1,\ldots ,uX'_n\), where the two sequences together consist of independent copies of \(X\). Then

Let \(1\le n\le N\) and let \(W_K,W_Q\in M_{n\times N}(\mathbb {R})\) have jointly independent centered Gaussian entries, iid within each matrix, with respective nonzero variances \(v_K,v_Q\). For \(L=W_K^TW_Q\),

independently of the head dimension \(n\).

Let \(L\ne 0\) and \(\pi (L)=(a,b,c,d)\). The coordinates are nonnegative and sum to one. Their faces and vertices have the following fibers.

1. \(a=0\) exactly when \(L\) is symmetric, and \(a=1\) exactly when \(L\) is antisymmetric.

2. \(b=0\) exactly when \(S\) is semidefinite, and \(b=1\) exactly when \(L\) is symmetric and \(\alpha =\beta \).

3. \(d=0\) exactly when \(\alpha _j\ge \beta _j\) for every \(j\), equivalently \(S=H+P\) with \(H\) symmetric with spectrum symmetric about zero and \(P\) positive semidefinite. The matrices may be chosen to commute with \(S\) and with each other, with \(\lVert P\rVert ^2=c\lVert L\rVert ^2\). Also, \(d=1\) exactly when \(L\) is symmetric negative semidefinite.

4. \(c=0\) exactly when \(\beta _j\ge \alpha _j\) for every \(j\), equivalently \(S=H-P\) with the same conditions on \(H,P\). They may be chosen to commute with \(S\) and with each other, with \(\lVert P\rVert ^2=d\lVert L\rVert ^2\). Also, \(c=1\) exactly when \(L\) is symmetric positive semidefinite.